# geom2arith

Geometric to arithmetic moments of asset returns

## Syntax

``[ma,Ca = geom2arith(mg,Cg)``
``[ma,Ca = geom2arith(___,t)``

## Description

example

````[ma,Ca = geom2arith(mg,Cg)` transforms moments associated with a continuously compounded geometric Brownian motion into equivalent moments associated with a simple Brownian motion with a possible change in periodicity. ```

example

````[ma,Ca = geom2arith(___,t)` adds an optional argument `t`. ```

## Examples

collapse all

This example shows several variations of using `geom2arith`.

Given geometric mean `m` and covariance `C` of monthly total returns, obtain annual arithmetic mean `ma` and covariance `Ca`. In this case, the output period (1 year) is 12 times the input period (1 month) so that the optional input `t` = `12`.

```m = [ 0.05; 0.1; 0.12; 0.18 ]; C = [ 0.0064 0.00408 0.00192 0; 0.00408 0.0289 0.0204 0.0119; 0.00192 0.0204 0.0576 0.0336; 0 0.0119 0.0336 0.1225 ]; [ma, Ca] = geom2arith(m, C, 12)```
```ma = 4×1 0.5508 1.0021 1.0906 1.4802 ```
```Ca = 4×4 0.0695 0.0423 0.0196 0 0.0423 0.2832 0.1971 0.1095 0.0196 0.1971 0.5387 0.3013 0 0.1095 0.3013 1.0118 ```

Given annual geometric mean `m` and covariance `C` of asset returns, obtain monthly arithmetic mean `ma` and covariance `Ca`. In this case, the output period (1 month) is 1/12 times the input period (1 year) so that the optional input `t` = `1/12`.

`[ma, Ca] = geom2arith(m, C, 1/12)`
```ma = 4×1 0.0038 0.0070 0.0076 0.0103 ```
```Ca = 4×4 0.0005 0.0003 0.0001 0 0.0003 0.0020 0.0014 0.0008 0.0001 0.0014 0.0037 0.0021 0 0.0008 0.0021 0.0070 ```

Given geometric mean `m` and covariance `C` of monthly total returns, obtain quarterly arithmetic return moments. In this case, the output is `3` of the input periods so that the optional input `t` = `3`.

`[ma, Ca] = geom2arith(m, C, 3)`
```ma = 4×1 0.1377 0.2505 0.2726 0.3701 ```
```Ca = 4×4 0.0174 0.0106 0.0049 0 0.0106 0.0708 0.0493 0.0274 0.0049 0.0493 0.1347 0.0753 0 0.0274 0.0753 0.2530 ```

## Input Arguments

collapse all

Continuously compounded or geometric mean of asset returns, specified as an n-vector.

Data Types: `double`

Continuously compounded or geometric covariance of asset returns, specified as an `n`-by-`n` symmetric, positive semidefinite matrix. If `Cg` is not a symmetric positive semidefinite matrix, use `nearcorr` to create a positive semidefinite matrix for a correlation matrix.

Data Types: `double`

(Optional) Target period of geometric moments in terms of periodicity of arithmetic moments, specified as a scalar positive numeric.

Data Types: `double`

## Output Arguments

collapse all

Arithmetic mean of asset returns over the target period (`t`), returned as an n-vector.

Arithmetic covariance of asset returns over the target period (`t`), returned as an `n`-by-`n` matrix.

## Algorithms

Geometric returns over period tG are modeled as multivariate lognormal random variables with moments

`$E\left[\text{Y}\right]=1+{\text{m}}_{G}$`

and

`$\mathrm{cov}\left(\text{Y}\right)={\text{C}}_{G}$`

Arithmetic returns over period tA are modeled as multivariate normal random variables with moments

`$E\left[\text{X}\right]={\text{m}}_{A}$`

`$\mathrm{cov}\left(\text{X}\right)={\text{C}}_{A}$`

Given t = tA / tG, the transformation from geometric to arithmetic moments is

`${\text{C}}_{{A}_{ij}}=t\mathrm{log}\left(1+\frac{{\text{C}}_{{G}_{ij}}}{\left(1+{\text{m}}_{{G}_{i}}\right)\left(1+{\text{m}}_{{G}_{j}}\right)}\right)$`

`${\text{m}}_{{A}_{i}}=t\mathrm{log}\left(1+{\text{m}}_{{G}_{i}}\right)-\frac{1}{2}{\text{C}}_{{A}_{ii}}$`

For i,j = 1,..., n.

Note

If t = 1, then X = log(Y).

This function requires that the input mean must satisfy ```1 + mg > 0``` and that the input covariance `Cg` must be a symmetric, positive, semidefinite matrix.

The functions `geom2arith` and `arith2geom` are complementary so that, given `m`, `C`, and `t`, the sequence

```[ma,Ca] = geom2arith(m,C,t); [mg,Cg] = arith2geom(ma,Ca,1/t);```

yields `mg` = `m` and `Cg` = `C`.